How does thermal conductivity of metals decrease with the increase in temperature?

Hint: We know that the thermal conductivity of a material is generally defined as a measure of its ability to conduct heat. When heat is transferred, it takes place at a lower rate in case of low thermal conductivity materials whereas it takes place at a higher rate in materials of high thermal conductivity.

Complete answer:
Let us first understand about thermal conductivity as follows:-
-Thermal conductivity: It can be referred to as the measure of material’s ability to conduct or transfer heat. When heat is transferred, it takes place at a lower rate in materials of low thermal conductivity whereas it takes place at a higher rate in materials of high thermal conductivity.
-The thermal conductivity of any material mainly depends on following two things:-
(a) Motion of electrons in the material
(b) Molecular vibrations
-Usually in the case of metals, the thermal conductivity is mainly the function of motion of the free electrons. When the temperature of the metal is increased, the molecular vibrations of the material increases and this in turn decreases the mean free path of the electrons in it. So these molecular vibrations obstruct the flow of free electrons with the increment in temperature and result in reducing the conductivity of heat in metals.
-Therefore we can say that the thermal conductivity of solid metals decreases with the increase in temperature due to the obstruction in the flow of free electrons.

Note:
-Always remember that in case of non-metals, there are usually no free electrons (except for graphite which is a good conductor of heat and electricity) and therefore only the molecular vibrations are responsible for conduction of heat. So we can say that the conductivity of non-metals increases as we increase the temperature due to increase in molecular vibrations.